Scalar curvature equation on warped products manifolds

In this aritcle by Dobarro and Dozo, the authors give another method calculating the scalar curvatrue on warped products manifolds by making using of conformal change.

Let M = (M^m,g) and N = (N^n,h) be two Riemann manifolds. For \phi\in C^{\infty}(M), \phi> 0 on M, we consider the warped product

\displaystyle M\times_{\phi} N = ((M\times N)^{m+n}, g + \phi^2 h)

and show the relationship between the scalar curvatures R_g on M, R_h on N and R_{\phi} on M \times_{\phi} N. This relationship is a nonlinear partial differential equation satisfied by a power of the weight \phi.

Theorem:  Let R_g, R_h and R_{\phi} denote the scalar curvature on M, N and M \times_{\phi} N respectively. Then the following equality holds:

\displaystyle  -\frac{4n}{n+1}\Delta_gu+R_gu+R_hu^{\frac{n-3}{n+1}}=R_{\phi}u\hfill (1)

where u=\phi^{\frac{n+1}2}. Namely,

\displaystyle  R_{\phi}=R_g+\frac{R_h}{\phi^2}-2n\frac{\Delta\phi}{\phi}-n(n-1)\frac{|\nabla_g\phi|^2}{\phi^2}.

Proof: Write g_{\phi}\doteqdot g+\phi^2h=\phi^2(\phi^{-2}g+h), so g_{\phi} is conformal to \tilde{g}_{\phi}\doteqdot\phi^{-2}g+h on M\times N and \phi^{-2}g is conformal to g on M.

Suppose m \ge 3, we apply Yamabe equation in M to obtain that \phi satisfies

\displaystyle  -\frac{4(m-1)}{m-2}\Delta_gv+R_gv=v^{\frac{n+2}{n-2}}\widetilde{R}_g\hfill (2)

with \phi^2=v^{\frac4{m-2}}, where \widetilde{R}_g denotes the scalar curvature on (M^m,\phi^{-2}g).
As m+n\ge3, we use Yamabe equation in M\times N. Hence \phi also satisfies

\displaystyle  -\frac{4(m+n-1)}{m+n-2}\Delta_{\tilde{g}_{\phi}}w+\widetilde{R}_{g_{\phi}}w={R}_{g_{\phi}}w^{\frac{m+n+2}{m+n-2}}\hfill (3)

with w^{\frac4{m+n-2}}=\phi^2 where \widetilde{R}_{g_{\phi}} denotes the scalar curvature on (M^m+N^n,\tilde{g}_{\phi}) and \Delta_{\tilde{g}_{\phi}} the corresponding laplacian.

From w\in C^{\infty}(M), we deduce that

\displaystyle  \Delta_{\tilde{g}_{\phi}}w=\Delta_{\phi^{-2}g+h}w=\Delta_{\phi^{-2}g}w.

Working in local coordinates

\displaystyle  \Delta_{\phi^{-2}g}w=\frac1{\sqrt{|\phi^{-2}g|}}\partial_i\left((\phi^{-2}g)^{ij}\sqrt{|\phi^{-2}g|}\partial_jw\right)

where |\phi^{-2}g|=\det(\phi^{-2}g_{ij})=\phi^{-2m}|g| and (\phi^{-2}g)^{ij}=\phi^2g^{ij}, hence

\displaystyle\begin{gathered}  \Delta_{\phi^{-2}g}w=\phi^m\frac1{\sqrt{|g|}}\partial_i\left(\phi^{2-m}g^{ij}\sqrt{|g|}\partial_jw\right)\hfill\\  \qquad=v^{\frac{-2m}{m-2}}\frac1{\sqrt{|g|}}\partial_i\left(v^2g^{ij}\sqrt{|g|}\partial_jw\right)\hfill\\  \qquad=\left(v\Delta_gw+2g^{ij}\partial_iv\partial_jw\right)v^{-\frac{n+2}{n-2}}.  \end{gathered}

On the other hand,

\displaystyle  \Delta_g(vw)=v\Delta_gw+w\Delta_gv+2g^{ij}\partial_iv\partial_jw.

Hence

\displaystyle  v^{\frac{n+2}{n-2}}\Delta_{\phi^{-2}g}w=\Delta_g(vw)-w\Delta_gv.\hfill(4)

We have

\displaystyle  \widetilde{R}_{g_{\phi}}=\widetilde{R}_g+R_h\hfill(5)

By using of this in (3)  and multiplying by v^{\frac{n+2}{n-2}}, we obtain from (4)

\displaystyle  -\frac{4(m+n-1)}{m+n-2}\big(\Delta_g(vw)-w\Delta_gv\big)+v^{\frac{n+2}{n-2}}(\widetilde{R}_g+R_h)={R}_{g_{\phi}}w^{\frac{m+n+2}{m+n-2}}v^{\frac{n+2}{n-2}}\hfill (6)

From (2) we arrive at

\displaystyle\begin{gathered}  -\frac{4(m+n-1)}{m+n-2}\Delta_g(vw)-\frac{4n}{(m+n-2)(m-2)}w\Delta_gv\hfill\\  \qquad+R_g(vw)+R_hwv^{\frac{n+2}{n-2}}=R_{\phi}(vw)w^{\frac4{m+n-2}}v^{\frac4{m-4}}=R_{\phi}(vw).  \end{gathered}

Recalling that w^{\frac4{m+n-2}}v^{\frac4{m-4}}=\phi^2\phi^{-2}=1 and denoting u =\phi^{\frac{n+1}2}, replacing in terms of u in this last equality, then multiplying by u^{1/n+1}, we obtain

\displaystyle\begin{gathered}  -\frac{4(m+n-1)}{m+n-2}\Delta_g(u^{\frac n{n+1}})u^{\frac1{n+1}}-\frac{4n}{(m+n-2)(m-2)}u^{\frac{m+n-1}{n+1}}\Delta_gu^{\frac{2-m}{n+1}}\hfill(7)\\  \quad+R_gu+R_hu^{\frac{n-3}{n+1}}=R_{\phi}u.  \end{gathered}

For any \alpha\neq0, u\in C^{\infty}(M), u>0 such that

\displaystyle  \Delta_gu^{\alpha}=\alpha(\alpha-1)u^{\alpha-2}|\nabla_gu|^2+\alpha u^{\alpha-1}\Delta_gu.

Choosing \alpha=\frac n{n+1} and \frac{2-m}{n+1} in (7), we obtain

\displaystyle  -\frac{4n}{n+1}\Delta_gu+R_gu+R_hu^{\frac{n-3}{n+1}}=R_{\phi}u.

Advertisements
此条目发表在Analysis, Geometry分类目录,贴了, , , 标签。将固定链接加入收藏夹。

发表评论

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / 更改 )

Twitter picture

You are commenting using your Twitter account. Log Out / 更改 )

Facebook photo

You are commenting using your Facebook account. Log Out / 更改 )

Google+ photo

You are commenting using your Google+ account. Log Out / 更改 )

Connecting to %s